What is 11.1612121212... as a Fraction?

11.1612 equals 9208825 as a fraction.

How do you turn 11.1612 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1612 repeating into a fraction, begin writing this simple equation:

n = 11.1612 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁴ = 10000.

10000 × n = 111612.12 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

10000 × n = 111612.12
  100 × n = 11.1612
 9900 × n = 110496

1104969900 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 12 (the GCF - greatest common factor).

n = 1104969900 = 110496 ÷ 129900 ÷ 12 = 9208825. So,

11.1612 = 9208825 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 9208825 is also equal to 11133825 when expressed as a mixed number.

The repeating decimal 11.1612 (vinculum notation) has a repeated block length of 2. It is also represented as 11.16121212... (ellipsis notation) which equals approximately 11.161212121212 (decimal approximation)(*).

The recurring decimal 11.1612 can be written as a ratio of two integers having 9208 as the numerator and 825 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.